{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:5OFDPJPMTGIQDFJMKASOVKZEJW","short_pith_number":"pith:5OFDPJPM","canonical_record":{"source":{"id":"1004.4407","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-26T03:56:22Z","cross_cats_sorted":["math.CA","math.SP"],"title_canon_sha256":"a47f7529c297ca4a00c0f5e33993aefa85aaa56c814b353f3b48cf1c79bfee55","abstract_canon_sha256":"05e7022af2e39ed62ee395c7a293998c5c6b0c3c17e0e8953350482b17e7e963"},"schema_version":"1.0"},"canonical_sha256":"eb8a37a5ec999101952c5024eaab244d933d1f42e1375f4f8fbbe19fa64f7244","source":{"kind":"arxiv","id":"1004.4407","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.4407","created_at":"2026-05-18T04:41:36Z"},{"alias_kind":"arxiv_version","alias_value":"1004.4407v4","created_at":"2026-05-18T04:41:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4407","created_at":"2026-05-18T04:41:36Z"},{"alias_kind":"pith_short_12","alias_value":"5OFDPJPMTGIQ","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5OFDPJPMTGIQDFJM","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5OFDPJPM","created_at":"2026-05-18T12:26:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:5OFDPJPMTGIQDFJMKASOVKZEJW","target":"record","payload":{"canonical_record":{"source":{"id":"1004.4407","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-26T03:56:22Z","cross_cats_sorted":["math.CA","math.SP"],"title_canon_sha256":"a47f7529c297ca4a00c0f5e33993aefa85aaa56c814b353f3b48cf1c79bfee55","abstract_canon_sha256":"05e7022af2e39ed62ee395c7a293998c5c6b0c3c17e0e8953350482b17e7e963"},"schema_version":"1.0"},"canonical_sha256":"eb8a37a5ec999101952c5024eaab244d933d1f42e1375f4f8fbbe19fa64f7244","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:41:36.645523Z","signature_b64":"wh/74c8473MIyKdYoA5PknkvkqVMboiugGv87GD9XAu0+HzuRYCfYIeThd/7dO9qo6jCJbdNzPv/k45bSUPZAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb8a37a5ec999101952c5024eaab244d933d1f42e1375f4f8fbbe19fa64f7244","last_reissued_at":"2026-05-18T04:41:36.645079Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:41:36.645079Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1004.4407","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4QsWb9nNFvmudfARE8fYzwgoKdXWJu2YZLT5fD3j3/DP6ysMQtc5tJvuSzvF6F99glUJ6ZMeULrFcRUb4V89DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T18:41:26.095732Z"},"content_sha256":"3b2909b64de6ef1b44b8168d73ce3a661254f222f6da8b0a78e71c6ec9f4fa00","schema_version":"1.0","event_id":"sha256:3b2909b64de6ef1b44b8168d73ce3a661254f222f6da8b0a78e71c6ec9f4fa00"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:5OFDPJPMTGIQDFJMKASOVKZEJW","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Speed of stability for birth--death processes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.SP"],"primary_cat":"math.PR","authors_text":"Mu-Fa chen","submitted_at":"2010-04-26T03:56:22Z","abstract_excerpt":"This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or having general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which the criterion for the positivity of the rate and an approxi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4407","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:41:36Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YBcWQztZgQyeGOvnZLCoezI89BjYTRE68f6cEqjkWyXwV7AHALhj25GX1Nz8jdcz2wsNdHR4QLMH4WvsYS0YDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-23T18:41:26.096107Z"},"content_sha256":"24feb63656db7258f44f1b9c1b5ec2221337640e814ac0fc50ad2f403890fe09","schema_version":"1.0","event_id":"sha256:24feb63656db7258f44f1b9c1b5ec2221337640e814ac0fc50ad2f403890fe09"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5OFDPJPMTGIQDFJMKASOVKZEJW/bundle.json","state_url":"https://pith.science/pith/5OFDPJPMTGIQDFJMKASOVKZEJW/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5OFDPJPMTGIQDFJMKASOVKZEJW/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-23T18:41:26Z","links":{"resolver":"https://pith.science/pith/5OFDPJPMTGIQDFJMKASOVKZEJW","bundle":"https://pith.science/pith/5OFDPJPMTGIQDFJMKASOVKZEJW/bundle.json","state":"https://pith.science/pith/5OFDPJPMTGIQDFJMKASOVKZEJW/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5OFDPJPMTGIQDFJMKASOVKZEJW/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:5OFDPJPMTGIQDFJMKASOVKZEJW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"05e7022af2e39ed62ee395c7a293998c5c6b0c3c17e0e8953350482b17e7e963","cross_cats_sorted":["math.CA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-26T03:56:22Z","title_canon_sha256":"a47f7529c297ca4a00c0f5e33993aefa85aaa56c814b353f3b48cf1c79bfee55"},"schema_version":"1.0","source":{"id":"1004.4407","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1004.4407","created_at":"2026-05-18T04:41:36Z"},{"alias_kind":"arxiv_version","alias_value":"1004.4407v4","created_at":"2026-05-18T04:41:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.4407","created_at":"2026-05-18T04:41:36Z"},{"alias_kind":"pith_short_12","alias_value":"5OFDPJPMTGIQ","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_16","alias_value":"5OFDPJPMTGIQDFJM","created_at":"2026-05-18T12:26:04Z"},{"alias_kind":"pith_short_8","alias_value":"5OFDPJPM","created_at":"2026-05-18T12:26:04Z"}],"graph_snapshots":[{"event_id":"sha256:24feb63656db7258f44f1b9c1b5ec2221337640e814ac0fc50ad2f403890fe09","target":"graph","created_at":"2026-05-18T04:41:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper is a continuation of the study on the stability speed for Markov processes. It extends the previous study of the ergodic convergence speed to the non-ergodic one, in which the processes are even allowed to be explosive or having general killings. At the beginning stage, this paper is concentrated on the birth-death processes. According to the classification of the boundaries, there are four cases plus one having general killings. In each case, some dual variational formulas for the convergence rate are presented, from which the criterion for the positivity of the rate and an approxi","authors_text":"Mu-Fa chen","cross_cats":["math.CA","math.SP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-26T03:56:22Z","title":"Speed of stability for birth--death processes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.4407","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:3b2909b64de6ef1b44b8168d73ce3a661254f222f6da8b0a78e71c6ec9f4fa00","target":"record","created_at":"2026-05-18T04:41:36Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"05e7022af2e39ed62ee395c7a293998c5c6b0c3c17e0e8953350482b17e7e963","cross_cats_sorted":["math.CA","math.SP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2010-04-26T03:56:22Z","title_canon_sha256":"a47f7529c297ca4a00c0f5e33993aefa85aaa56c814b353f3b48cf1c79bfee55"},"schema_version":"1.0","source":{"id":"1004.4407","kind":"arxiv","version":4}},"canonical_sha256":"eb8a37a5ec999101952c5024eaab244d933d1f42e1375f4f8fbbe19fa64f7244","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb8a37a5ec999101952c5024eaab244d933d1f42e1375f4f8fbbe19fa64f7244","first_computed_at":"2026-05-18T04:41:36.645079Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:41:36.645079Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wh/74c8473MIyKdYoA5PknkvkqVMboiugGv87GD9XAu0+HzuRYCfYIeThd/7dO9qo6jCJbdNzPv/k45bSUPZAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:41:36.645523Z","signed_message":"canonical_sha256_bytes"},"source_id":"1004.4407","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3b2909b64de6ef1b44b8168d73ce3a661254f222f6da8b0a78e71c6ec9f4fa00","sha256:24feb63656db7258f44f1b9c1b5ec2221337640e814ac0fc50ad2f403890fe09"],"state_sha256":"39ddcf0c00673bf544c21d00a44a907dae85c7e73237a3714f71381fdc7f507d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"POIFf55LJ7Qx8EwjpLFwi4lnJV1uD/JSuR1iPIY9kVKPIxYHB5zfmGRPqQ/hUX0CeSWCcJ7o6gC7rRISY2pPAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-23T18:41:26.098943Z","bundle_sha256":"be2878a457b54c510b3a72807ea98fadece16658b7af1cb99e951949378a8be7"}}